06 - 逆矩陣、列空間與零空間 · b站_線性代數的本質

求矩陣的行列式，矩陣的行和列要相等 ***** 矩陣可以用來求解線性方程組： ![](https://img.kancloud.cn/94/f2/94f2efec17b2aeb055211dff65650d47_1326x623.png) 一個線性變換將向量x變為向量v ![](https://img.kancloud.cn/81/a0/81a05d7e1607291a78857cd0954e49d0_1192x591.png) ***** ![](https://img.kancloud.cn/76/f7/76f7c03e8faf682ed9386d7c0a69ae4b_1083x693.png) ![](https://img.kancloud.cn/db/29/db29086ff9d40d375842d953f7be752e_1279x729.png) ![](https://img.kancloud.cn/8b/68/8b68f76a423fe9729c148a074493ea06_1092x706.png) ![](https://img.kancloud.cn/1b/82/1b823ff49e934a10f42d399bc9c39a14_1215x693.png) ![](https://img.kancloud.cn/36/da/36da8a1f1d9fe31b358abd69d77c2771_1119x495.png) ***** 變換A的行列式不為0 ![](https://img.kancloud.cn/b1/2b/b12bed625402732d24ed109e91264560_1083x700.png) v向量通過逆變換來找到x向量 ![](https://img.kancloud.cn/91/6d/916d106e6809a1ea43a802cf6e206606_974x706.png) **逆變換** ![](https://img.kancloud.cn/11/e1/11e1de0a94c5350f80bcbad96bbc3754_1331x731.png) 先進行線性變換A，再進行逆變換。結果為原坐標系回到初始狀態，基向量**i**和**j**都保持不變 ![](https://img.kancloud.cn/a1/ac/a1acd526ccb931a1c0e091cdee7d138a_1302x740.png) ***** ![](https://img.kancloud.cn/bd/05/bd05028d3cfc439689decee7475f8765_959x608.png) 在兩邊同時乘A的逆矩陣來求解向量X。在幾何上的意義就是對向量v進行A的逆向變換來找到向量x ![](https://img.kancloud.cn/fe/41/fe4122f5cc198a26418b2e5dedab91a9_921x536.png) ***** ## 當行列式det(A)=0時 ![](https://img.kancloud.cn/d8/90/d8907b51a7fcef5142a60bd3f8ce9f52_1262x702.png) ![](https://img.kancloud.cn/61/c7/61c7ded1069453dc1766bde00fe9a629_1197x727.png) ![](https://img.kancloud.cn/8f/31/8f3186256cbc552568474b1e03dfb1a3_1213x711.png) ![](https://img.kancloud.cn/f9/3b/f93bfc1c219308217e580756ec6c2187_1288x709.png) ![](https://img.kancloud.cn/3c/ee/3cee73633e710019cd08a578deeb5b5b_1288x764.png) ![](https://img.kancloud.cn/c8/cf/c8cf22287deaf7a331e8432ca212996a_1332x727.png) ***** ![](https://img.kancloud.cn/b0/77/b07706c8dfdd99d9cd8ee25993cb6725_1302x714.png) ***** ![](https://img.kancloud.cn/56/35/56351c6a1b0af9108e90d56209a8591f_1255x721.png) ![](https://img.kancloud.cn/b1/f8/b1f8f8d7d30b2383ca3dc7609883b80b_1173x694.png) ***** 當變換將二維壓縮成一條直線，稱這個變換的秩為1 ![](https://img.kancloud.cn/f4/6f/f46f1d9994b2d299364663a7b1652b54_1324x704.png) ![](https://img.kancloud.cn/b2/b1/b2b17fed193c9dd27397657fc4920455_1265x706.png) ![](https://img.kancloud.cn/e1/61/e161741304310659f4e14e8fec2314e0_1267x707.png) ![](https://img.kancloud.cn/61/75/61756dd0752f215732353781930ec1ed_981x645.png) ***** 基向量可以張成二維空間，且矩陣的行列式的不為零 ![](https://img.kancloud.cn/72/a3/72a349b5d838f4b4e1ba6635c75496be_1269x693.png) ***** ![](https://img.kancloud.cn/73/36/7336b99456d8eb732a2ca467f86bd85c_1207x703.png) ***** 所有矩陣變換變換結果的集合是矩陣的"列空間" ![](https://img.kancloud.cn/e2/c1/e2c1fc107f69e9d3f955c72073599fa0_1157x698.png) **原因** ![](https://img.kancloud.cn/98/4a/984a17f4c5ecac3d6add14b1d9121806_1278x696.png) ![](https://img.kancloud.cn/7c/12/7c12fd531438e85f5fea4f2484ea3ca6_1187x703.png) ![](https://img.kancloud.cn/f3/e4/f3e4ec836da99a89fcc1e13c710a6018_1165x697.png) **秩的精確定義** ![](https://img.kancloud.cn/ae/77/ae7799a18b50ef56b3935944a152ba2e_614x93.png) ![](https://img.kancloud.cn/73/c0/73c0bbf19b14c725c3f4147f0a0ac133_626x97.png) ![](https://img.kancloud.cn/e4/30/e4301e18dd2a72446d4cdfeb9322a9b5_1195x704.png) ![](https://img.kancloud.cn/f7/c6/f7c6150145261ee17c3c24a661c06987_887x711.png) 因為線性變換原點不變 ![](https://img.kancloud.cn/70/68/7068264964635bfcc2fc2e6dc1041a7b_1096x727.png) ***** ![](https://img.kancloud.cn/9c/8e/9c8ef6620eca2aa7d7623e94b6f71025_1128x708.png) ![](https://img.kancloud.cn/72/ba/72ba11d72116869878a7f58c624eef37_1094x719.png) ***** ![](https://img.kancloud.cn/f7/eb/f7eb2727b4f2e94ced5c19c2b456961c_1046x700.png) ![](https://img.kancloud.cn/c4/03/c403e1b567c01420fc65d2c80a825421_1169x703.png) ![](https://img.kancloud.cn/3d/09/3d09c3c5778afc0670165681aff306ae_1169x667.png) ![](https://img.kancloud.cn/3e/e3/3ee37fb6cad5e14fcdbe1f4897ad1a41_1049x681.png) ***** ![](https://img.kancloud.cn/41/a9/41a91c8ff95f0e3638c1881dd4c48fcc_927x658.png) ![](https://img.kancloud.cn/d5/e8/d5e84bde14d521f8e61e314e3b34f9f6_937x698.png)